The matrix Heinz mean and related divergence

Abstract

In this paper, we introduce a new quantum divergence
$$\Phi (X,Y) = \Tr \left[\left(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right)X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$
where $0< \alpha <1$.
We study the least square problem with respect to this divergence. We also show that the new quantum divergence satisfies the Data Processing Inequality in quantum information theory. In addition, we show that the matrix $p$-power mean $\mu_p(t, A, B) = ((1-t)A^p + tB^p)^{1/p}$ satisfies the in-betweenness property with respect to the new divergence.

Keywords

• Quantum divergence
• Heinz mean
• least squares problems
• matrix power mean
• in-betweenness property
• data processing inequality

References

1. [1] S. Amari, Information Geometry and Its Applications. Springer, 2016.
2. [2] K.M.R. Audenaert, In-betweenness, a geometrical monotonicity property for operator means, Linear Algebra Appl. 438, 1769-1778, 2013.
3. [3] R. Bhatia, S. Gaubert and T. Jain, Matrix versions of Hellinger distance, Lett. Math. Phys. 109, 2779-2781, 2019.
4. [4] T.H. Dinh, R. Dumitru and J.A. Franco, On the monotonicity of weighted power means for matrices, Linear Algebra Appl. 527, 128-140, 2017.
5. [5] T.H. Dinh, R. Dumitru and J.A. Franco, Some geometric properties of matrix means in different distance functions, Positivity, 24, 1419-1434, 2020.
6. [6] T.H. Dinh, B.K. Vo and T.Y. Tam, In-sphere property and reverse inequalities for matrix means, Electron. J. Linear Algebra, 35 (1), 35-41, 2019.
7. [7] T.H. Dinh, C.T. Le, B.K. Vo and T.D. Vuong, Weighted Hellinger Distance and In-betweenness property, Math. Inequal. Appl. 24 (1), 157-165, 2021.
8. [8] T.H. Dinh, C.T. Le, B.K. Vo and T.D. Vuong, The $\alpha$-z-Bures Wesserstein divergence, Linear Algebra Appl. 624, 267-280, 2021.

9. [9] F. Franco and R. Dumitru, Generalized Hellinger Metrics and Audenaert’s In- Betweenness, Linear Algebra Appl. 585, 191-198, 2020.
10. [10] F. Frank and E. Lieb, Monotonicity of a relative Renyi entropy, J. Math. Phys. 54, 122201, 2013.
11. [11] F. Hiai and D. Petz, Introduction to Matrix Analysis and Application, Springer, 2014.
12. [12] N. Lam and P.L. Le, Quantum divergences with p-power means, Linear Algebra Appl. 609, 289-307, 2021.
13. [13] N. Lam and R. Milley, Some notes on quantum Hellinger divergences with Heinz means, Electron. J. Linear Algebra, 36, 704-722, 2020.
14. [14] M.M. Wolf, Quantum channels and operations: Guided tour, Lecture Notes Avail- able at http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/ QChannelLecture.pdf, 2012.
15. [15] J. Pitrik and D. Virosztek, A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means, Linear Algebra Appl. 609, 203-217, 2021.